Actual Infinity in Reality?

[SteveII]

I set up four such thought experiments above. We clearly have contradictions that arise when comparing these possible worlds.

You are once again conflating the ability or inability to imagine something as being the same as demonstrating that something is or is not logically possible.  It is the strength of imagination that we have the ability to conceive of impossible things.  I can imagine that there is a possible world where God does not exist.  Have I thus demonstrated that God is not a necessary being?  No, I have not.  If God is a necessary being and I imagine that God does not exist in a possible world, all I've shown is that my imagination is at odds with my assumptions.  Your thought experiments don't add anything to the assumptions and conclusions you had prior to the thought experiment.

As long as we're on the subject though, allow me to make several notes,

1. Infinity, while treated as a number, is not a number in the sense that the counting numbers are.  Therefore the equations you are presenting above have to be construed as set theoretic operations.  As such, there is nothing contradictory about the set theoretic results.  It only appears that way if you are construing the equations as normal numerical operations.  Thus presenting the equations adds nothing and seems to serve only to mislead.

2. From what I understand of possible worlds semantics, the idea of comparing one possible world to another, different possible world is not supported.  If you think it is, then I'd request that you show which possible world semantics you are referencing.  If you can't compare possible worlds meaningfully, then attempting to even formulate Hilbert's hotel's operation in terms of possible world semantics is not possible.

3. Hilbert's hotel applies to sets that are countably infinite.  If time is continuous and infinite, it would seem that the set of all possible moments is uncountably infinite.  In that event, Hilbert's hotel simply wouldn't apply.  As long as we're throwing around burden of proof questions, I think you are obligated to either show that time is not continuous, or that even if it is, that the set of all possible moments is a countable infinity.  Otherwise, we can simply dispense with Hilbert's hotel, as it does not cover all the possibilities for a temporally infinite universe that I have raised.  An objection which only applies to some of the possibilities but not all cannot possibly demonstrate that all cases are impossible.

But the real problem is that your position is that all of them are true despite the obvious contradictions. You have not shown why we should accept the contradictions other than to wonder if that's just the way it is. It seems to me that you have some burden of proof to shoulder if you are proposing ignoring obvious contradictions. 

Since you haven't actually shown any such alleged contradictions, I have great difficulty making sense of your complaint here.  I'm supposed to refute the existence of contradictions you haven't demonstrated?  That's ballsy, but ridiculous.  I can't refute a case that you haven't made.  So, no, I don't assume any burden of proof to show that something you claim exists doesn't exist.  You need to first demonstrate the existence of these alleged contradictions.  Once you've shouldered your burden of proof, we'll see what obligations I have in return.

Why isn't that question begging? The proposition is that an actual infinity does not exist. To dismiss thought experiments on the basis they don't work with actual infinities needs a little more support.

Well, first of all, you're moving the goalpost.  The question is whether or not an actual infinity is logically possible.  The claim that the proposition is whether an actual infinity actually exists is asking me to demonstrate that a specific actual infinity is in fact actual.  Those are different standards.  I don't know that I could prove that time is temporally infinite even if I wanted to do so.  I never claimed as much.  Only that the idea of a temporal infinity is consistent, both logically, and with known models of physics and cosmology.  I believe I've done that.  Your job as my interlocutor is to show that I've missed a contradiction which exists.  In that context, I am suggesting that the so-called absurdity that results in the thought experiment may be a product of an incomplete set of intuitions about reference.  It's a possibility.  Your task, is to show that the absurdity in Hilbert's hotel is metaphysically real, not just a product of intuitional failure.  You so far have not done so, and continue to talk around the problem rather than addressing it.

It seems to me you are accepting an actual infinity as a brute fact. Can accepting a brute fact really be considered "logical".

No, as I just pointed out, I'm accepting that the hypothesis that time is temporally infinite is both logically and physically consistent.  But that doesn't seem to be your point here.  Your question as to whether what I'm doing is "logical" seems to be nothing more than a rhetorical smear.  If you're reduced to such smears, I have to question what you hope to achieve with it?  Treating something as a brute fact is neither logical nor illogical, so I can only assume that, instead, you are simply trying to suggest that I'm being irrational.  I don't see that as a productive path to a convincing argument.  It seems little more than an attempt to distract from the point I made, that you had not shown that any metaphysical assumption has been violated, and thereby avoid actually showing such a contradiction. 

Okay, let's back up to your initial position and try another tack. I think the best sentence that sums up your position is: " So, to the best I can tell, the idea that the universe is temporally infinite is consistent with a B theory of time and with some models of cosmology. So, ultimately, it doesn't appear that the case that you can't have an actual infinite has been made."

Under standard cosmology models, the B Theory of time has a beginning (at the time of the Bib Bang). An event creates a spacetime manifold. That at least make sense because we have a beginning and we are posit a potential infinite off into the future. By assuming an eternal universe model is correct, you assumed an eternal manifold and then...an actual infinite is possible.

After reading about the cosmology model you mentioned (CCC-Penrose). I noticed from your wiki link that "Penrose's basic construction[5] is to connect a countable sequence of open Friedmann–Lemaître–Robertson–Walker metric (FLRW) spacetimes, each representing a big bang followed by an infinite future expansion." This does not seem to be using the B Theory of time's manifold and simply claiming it never had a beginning (therefore an actual infinite). The theory proposes a "countable sequence" of different spacetime manifolds. Each manifold exists in sequence and therefore was never part of one big spacetime that existed as one eternal block.

So, it would seem that proposing the two theories together does not get you to even a model of an actual infinite and brings the question right back to, metaphysically speaking, can we have an actual infinite of past events? Since you can't get an eternal spacetime block out of any theory, you must have successive states of affairs. If you have successive states of affairs, they cannot be past infinite, because you will never get to our current state of affairs because an infinite number of prior states of affairs would have to happen.